Approximation Fixpoint Theory and the Semantics of Logic and Answers Set Programs

Approximation Fixpoint Theory was developed as a fixpoint theory of lattice operators that provides a uniform formalization of four main semantics of three major nonmonotonic reasoning formalisms. This paper clarifies how this fixpoint theory can define the stable and well-founded semantics of logic programs. It investigates the notion of strong equivalence underlying this semantics. It also shows the remarkable power of this theory for defining natural and elegant versions of these semantics for extensions of logic and answer set programs. In particular, we here consider extensions with general rule bodies, general interpretations (also non-Herbrand interpretations) and aggregates. We also investigate the relationship with the equilibrium semantics of nested answer set programs, on the formal and the informal level.

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